Set-Valued and Variational Analysis

, Volume 17, Issue 2, pp 153–182

A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory



It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre–Fenchel conjugates for set-valued functions is introduced and a Moreau–Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.


Set order relations Legendre–Fenchel conjugate Moreau–Fenchel theorem Set-valued function Conlinear space Set-valued risk measure 

Mathematics Subject Classifications (2000)

49N15 52A41 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA

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